Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data

Dedicated to Professor Kenji Yajima on his sixtieth birthday

Abstract

We study the asymptotic behavior in time of solutions to the Cauchy problem of nonlinear Schrödinger equations with a long-range dissipative nonlinearity given by λ |u|^p-1u in one space dimension, where 1 < p ≤ 3 (namely, p is a critical or subcritical exponent) and λ is a complex constant satisfying Im λ < 0 and ((p-1)/2√p) |Re λ| ≤ |Im λ|. We present the time decay estimates and the large-time asymptotics of the solution for arbitrarily large initial data, when “p = 3” or &ladquo;p < 3 and p is suitably close to 3”.